with \(\beta _0=(11C_A-2n_f)/(12\pi )\) . Note that the last equality holds with \(\alpha _\text{s} =\alpha _\text{s} (\mu _R)\)  at NLL accuracy.
We can use this to reflect the rapidity requirement given by the jet condition
by setting \(\eta _{\text{min},3} = \ln \mathopen {}\mathclose {\left(2\cosh (y_3)/R_0\right)}\) .
Here \(Q\)  is an arbitrary scale, which we choose to identify with the
partonic centre-of-mass energy, that ultimately cancels in the final expressions.
For global observables this cancellation happens with the soft function \(\mathcal {S}\) , however, the
restriction to radiation inside the jet additionally affects the
structure of soft emissions. The details were worked out in [1]}
for the jet-mass observable. Those results apply to the general class of
angularities as well, as they depend on the scaling of the observable with \(k_t\)  only, i.e. on the parameter \(a\)  in Eq. (REF ). This implies (using \(a=1\) )
\(\mathcal {S}^{\mathcal {B}_\delta }(L) = S_\text{global}^{\mathcal {B}_\delta }\mathopen {}\mathclose {\left(t(L)\right)}S_\text{non-global}^{\mathcal {B}_\delta }\mathopen {}\mathclose {\left(t(L)\right)}\;.\) 
